On The Nature Of Primes: A deterministic, endogenous, non-stationary S-adic Automaton for the sieve of Eratosthenes

We present a deterministic, endogenous, non-stationary S-adic automaton thatmodels the Sieve of Eratosthenes as a dynamical system over a finite symbolic alphabet. Theautomaton operates through three operators — shift, expansion, and filtering — applied sequentiallyto a growing symbolic tape, and provably reproduces the classical prime-compositeclassification for every integer n ≥ 2. Unlike algorithmic sieves, the automaton generatesan internal symbolic representation of the number line whose structure can be analyzed atevery step.Our first focus is: Can this new framework reproduce known mathematical knowledge?We demonstrate that this representation is not arbitrary: the tape exhibits a four-lettersubstructure {a, b, c, d} governed by an explicit substitution morphism and an upper triangulartransition matrix Mp. The dominant eigenvalue p−2 controls the population dynamicsof twin prime templates, yielding a recursive growth formula consistent with OEIS sequenceA059861 and with the combinatorial factors underlying the Hardy–Littlewood k-tuple conjecture.A central structural result is the Stability Zone SZn = [n+1, 2n−1], a provably immutableinterval in which prime candidates survive all prior filtering steps. Using a Frozen Windowtechnique, we verify the persistence of the symbolic structure experimentally up to n =250,000.Our second focus is: Can this new framework lead to new mathematical knowledge?Finally, we discuss the local Hausdorff–Besicovitch dimension of the prime candidateset within the Stability Zone. It begins near 0.92 and increases monotonically toward 1as n → ∞, following D(p) = ln(p − 1)/ ln(p). This process — vanishing fractality —unfolds dynamically inside the growing, advancing Stability Zone as it travels through thenumber line, and provides a deterministic, structural perspective on the transition from theordered regime of small primes to the apparent randomness observed in large-scale primedistributions.The automaton is offered not as a computational tool for generating primes, but as aresearch instrument: a symbolic framework in which arithmetic properties of the naturalnumbers emerge from the internal dynamics of the system.